The effects of perturbations on matrix-based signal processing

The effects of perturbations of the elements of a matrix on functions of the matrix, such as eigenvalues and eigenvectors, was a subject of great interest to applied mathematicians who were working in the early part of this century. For example, Rayleigh and others analyzed complicated linear systems by considering them to be perturbed versions of simpler, analytically tractable systems. Calculation of eigenvalue changes due to the perturbations were used to calculate corresponding changes in the natural frequencies of the system. This is of course very similar to recent work, except that in recent work the perturbations are usually assumed to be due to measurement noise.<>